Question

Consider a unity negative feedback control system with forward path gain: \[ G(s) = \frac{K}{(s+1)(s+2)(s+3)}. \] The impulse response of the closed-loop system decays faster than \(e^{-t}\) if \(\_\_\_\_\).

• A. \(1 \leq K \leq 5\)
• B. \(7 \leq K \leq 21\)
• C. \(-4 \leq K \leq -1\)
• D. \(-24 \leq K \leq -6\)

Hint:

To ensure a desired decay rate, analyze the real parts of the closed-loop poles using the characteristic equation and adjust \(K\) accordingly.

Answer 1

The Correct Option is A

Step 1: Derive the characteristic equation for the closed-loop system.
The closed-loop transfer function is given by: \[ T(s) = \frac{G(s)}{1 + G(s)} = \frac{\frac{K}{(s+1)(s+2)(s+3)}}{1 + \frac{K}{(s+1)(s+2)(s+3)}}. \] The characteristic equation of the system is: \[ (s+1)(s+2)(s+3) + K = 0. \] Step 2: Condition for faster decay than \(e^{-t}\).
For the impulse response to decay faster than \(e^{-t}\), all poles of the closed-loop system must have real parts less than \(-1\). Step 3: Determine the range of \(K\).
Solving the characteristic equation and analyzing the root locations, the condition \(1 \leq K \leq 5\) ensures that all poles of the system have real parts less than \(-1\). Final Answer: \[\boxed{{(1) } 1 \leq K \leq 5}\]